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* this work for additional information regarding copyright ownership.
* The ASF licenses this file to You under the Apache License, Version 2.0
* (the "License"); you may not use this file except in compliance with
* the License. You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
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package org.apache.commons.math3.analysis.solvers;
import org.apache.commons.math3.exception.NoBracketingException;
import org.apache.commons.math3.exception.NumberIsTooLargeException;
import org.apache.commons.math3.exception.TooManyEvaluationsException;
import org.apache.commons.math3.util.FastMath;
This class implements the
Muller's Method for root finding of real univariate functions. For
reference, see Elementary Numerical Analysis, ISBN 0070124477,
chapter 3.
Muller's method applies to both real and complex functions, but here we restrict ourselves to real functions. This class differs from MullerSolver in the way it avoids complex operations.
Except for the initial [min, max], it does not require bracketing
condition, e.g. f(x0), f(x1), f(x2) can have the same sign. If a complex
number arises in the computation, we simply use its modulus as a real
approximation.
Because the interval may not be bracketing, the bisection alternative is
not applicable here. However in practice our treatment usually works
well, especially near real zeroes where the imaginary part of the complex
approximation is often negligible.
The formulas here do not use divided differences directly.
See Also: Since: 1.2
/**
* This class implements the <a href="http://mathworld.wolfram.com/MullersMethod.html">
* Muller's Method</a> for root finding of real univariate functions. For
* reference, see <b>Elementary Numerical Analysis</b>, ISBN 0070124477,
* chapter 3.
* <p>
* Muller's method applies to both real and complex functions, but here we
* restrict ourselves to real functions.
* This class differs from {@link MullerSolver} in the way it avoids complex
* operations.</p><p>
* Except for the initial [min, max], it does not require bracketing
* condition, e.g. f(x0), f(x1), f(x2) can have the same sign. If a complex
* number arises in the computation, we simply use its modulus as a real
* approximation.</p>
* <p>
* Because the interval may not be bracketing, the bisection alternative is
* not applicable here. However in practice our treatment usually works
* well, especially near real zeroes where the imaginary part of the complex
* approximation is often negligible.</p>
* <p>
* The formulas here do not use divided differences directly.</p>
*
* @since 1.2
* @see MullerSolver
*/
public class MullerSolver2 extends AbstractUnivariateSolver {
Default absolute accuracy. /** Default absolute accuracy. */
private static final double DEFAULT_ABSOLUTE_ACCURACY = 1e-6;
Construct a solver with default accuracy (1e-6).
/**
* Construct a solver with default accuracy (1e-6).
*/
public MullerSolver2() {
this(DEFAULT_ABSOLUTE_ACCURACY);
}
Construct a solver.
Params: - absoluteAccuracy – Absolute accuracy.
/**
* Construct a solver.
*
* @param absoluteAccuracy Absolute accuracy.
*/
public MullerSolver2(double absoluteAccuracy) {
super(absoluteAccuracy);
}
Construct a solver.
Params: - relativeAccuracy – Relative accuracy.
- absoluteAccuracy – Absolute accuracy.
/**
* Construct a solver.
*
* @param relativeAccuracy Relative accuracy.
* @param absoluteAccuracy Absolute accuracy.
*/
public MullerSolver2(double relativeAccuracy,
double absoluteAccuracy) {
super(relativeAccuracy, absoluteAccuracy);
}
{@inheritDoc}
/**
* {@inheritDoc}
*/
@Override
protected double doSolve()
throws TooManyEvaluationsException,
NumberIsTooLargeException,
NoBracketingException {
final double min = getMin();
final double max = getMax();
verifyInterval(min, max);
final double relativeAccuracy = getRelativeAccuracy();
final double absoluteAccuracy = getAbsoluteAccuracy();
final double functionValueAccuracy = getFunctionValueAccuracy();
// x2 is the last root approximation
// x is the new approximation and new x2 for next round
// x0 < x1 < x2 does not hold here
double x0 = min;
double y0 = computeObjectiveValue(x0);
if (FastMath.abs(y0) < functionValueAccuracy) {
return x0;
}
double x1 = max;
double y1 = computeObjectiveValue(x1);
if (FastMath.abs(y1) < functionValueAccuracy) {
return x1;
}
if(y0 * y1 > 0) {
throw new NoBracketingException(x0, x1, y0, y1);
}
double x2 = 0.5 * (x0 + x1);
double y2 = computeObjectiveValue(x2);
double oldx = Double.POSITIVE_INFINITY;
while (true) {
// quadratic interpolation through x0, x1, x2
final double q = (x2 - x1) / (x1 - x0);
final double a = q * (y2 - (1 + q) * y1 + q * y0);
final double b = (2 * q + 1) * y2 - (1 + q) * (1 + q) * y1 + q * q * y0;
final double c = (1 + q) * y2;
final double delta = b * b - 4 * a * c;
double x;
final double denominator;
if (delta >= 0.0) {
// choose a denominator larger in magnitude
double dplus = b + FastMath.sqrt(delta);
double dminus = b - FastMath.sqrt(delta);
denominator = FastMath.abs(dplus) > FastMath.abs(dminus) ? dplus : dminus;
} else {
// take the modulus of (B +/- FastMath.sqrt(delta))
denominator = FastMath.sqrt(b * b - delta);
}
if (denominator != 0) {
x = x2 - 2.0 * c * (x2 - x1) / denominator;
// perturb x if it exactly coincides with x1 or x2
// the equality tests here are intentional
while (x == x1 || x == x2) {
x += absoluteAccuracy;
}
} else {
// extremely rare case, get a random number to skip it
x = min + FastMath.random() * (max - min);
oldx = Double.POSITIVE_INFINITY;
}
final double y = computeObjectiveValue(x);
// check for convergence
final double tolerance = FastMath.max(relativeAccuracy * FastMath.abs(x), absoluteAccuracy);
if (FastMath.abs(x - oldx) <= tolerance ||
FastMath.abs(y) <= functionValueAccuracy) {
return x;
}
// prepare the next iteration
x0 = x1;
y0 = y1;
x1 = x2;
y1 = y2;
x2 = x;
y2 = y;
oldx = x;
}
}
}